Oncogenic hyperplasia is the first and inevitable stage of formation of a (solid) tumor. This stage is also the core of many other proliferative diseases. The present work proposes the first minimal model that combines homeorhesis with oncogenic hyperplasia where the latter is regarded as a genotoxically activated homeorhetic dysfunction. This dysfunction is specified as the transitions of the fluid of cells from a fluid, homeorhetic state to a solid, hyperplastic-tumor state, and back. The key part of the model is a nonlinear reaction-diffusion equation (RDE) where the biochemical-reaction rate is generalized to the one in the wellknown Schlögl physical theory of the non-equilibrium phase transitions. A rigorous analysis of the stability and qualitative aspects of the model, where possible, are presented in detail. This is related to the spatially homogeneous case, i.e. when the above RDE is reduced to a nonlinear ordinary differential equation. The mentioned genotoxic activation is treated as a prevention of the quiescent G0-stage of the cell cycle implemented with the threshold mechanism that employs the critical concentration of the cellular fluid and the nonquiescentcell-duplication time. The continuous tumor morphogeny is described as a time-space-dependent cellular-fluid concentration. There are no sharp boundaries (i.e. no concentration jumps exist) between the domains of the homeorhesis-and tumor-cell populations. No presumption on the shape of a tumor is used. To estimate a tumor in specific quantities, the model provides the time-dependent tumor locus, volume, and boundary that also points out the tumor shape and size. The above features are indispensable in the quantitative development of antiproliferative drugs or therapies and strategies to prevent oncogenic hyperplasia in cancer and other proliferative diseases. The work proposes an analytical-numerical method for solving the aforementioned RDE. A few topics for future research are suggested.
As it follows from the results of C. H. Waddigton, F. E. Yates, A. S. Iberall, and other well-known bio-physicists, living fluids cannot be modelled within the frames of the fundamental assumptions of the statistical-mechanics formalism. One has to go beyond them. The present work does it by means of the generalized kinetics (GK), the theory enabling one to allow for the complex stochasticity of internal properties and parameters of the fluid particles. This is one of the key features which distinguish living fluids from the nonliving ones. It creates the disparity of the particles and hence breaks the eachfluid-component-uniformity requirement underlying statistical mechanics.The work deals with the corresponding modification of common kinetic equations which is in line with the GK theory and is the complement to the latter. This complement allows a subdivision of a fluid into the fluid components in terms of nondiscrete probability distributions. The treatment leads to one more equation that describes the above internal parameters. The resulting model is the system of these two equations. It appears to be always nonlinear in case of living fluids. In case of nonliving fluids, the model can be linear. Moreover, the living-fluid model, as a whole, cannot have the thermodynamic equilibrium, only partial equilibriums (such as the motional one) are possible. In contrast to this, in case of nonliving fluids, the thermodynamic equilibrium is, of course, possible. The number of the fluid components is treated as the number of the modes of the particle-characteristic probability density. In so doing, a fairly general extension of the notion of the mode from the one-dimensional case to the multidimensional case is proposed. The work also discusses the variety of the time-scales in a living fluid, the simplest quantum-mechanical equation relevant to living fluids, and the nonequilibrium nonlinear stochastic hydrodynamics option. The latter is simpler than, but conceptually comparable to, stochastic kinetic equations. A few directions for future research are suggested. The work notes a cohesion of mathematical physics and fluid mechanics with the living-fluid-related fields as a complex interdisciplinary problem.Keywords: Multicomponent fluid of living cells and related macromolecules; generalized kinetic theory; stochastic differential equation; probability distribution of parameters of the fluid particles; mode of probability distribution and multimodal distributions.
This work is devoted to the equilibrium distribution function for a fluid of mutually non-interacting identical composite point particles in three-dimensional physical space. The distribution function is derived within the generalized-kinetics (GK) vision from the proposed probabilistic model based on quantum-mechanical bosons and fermions. The first GK advantage is that the derivation does not involve any assumption on the interpolation between bosons and fermions whereas the resulting function provides this interpolation. The second GK advantage is that composons, the particles described with the GK-based distribution function, are considerably less schematic and more consistent physically than quons. Composons correspond to a specific case of Isakov's general q-commutation relation involving an infinite number of the q-coefficients. Connection of the composon concept to previous results in the literature is pointed out. A few directions for future research on the topic are formulated. The results of the work can be used in the composite-particle fluid problems where the Maxwell-Boltzmann description is not valid, for instance, in dense populations of not too massive point-like particles of a complex, composite nature at not too high temperatures.
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